Homomorphism from $A[X,Y]$ to $A[X]$ with kernel $(X^i-Y^j)$

Question

Let $A$ be an integral domain, $i,j \in \mathbb N$ such that gcd$(i,j)=1$. How would one define a homomorphism from $A[X,Y]$ to $A[X]$, having the ideal generated by $X^i-Y^j$ as its kernel?

Answer 1

This is really messy, especially since the $x$ in the target is not going to be the same as the $x$ in the domain, so I'll do this for

$$A[x,y]\to A[z]$$

for clarity.

Note that $x^i-y^j=0\iff x^i=y^{j}$. Fair enough, so then first we note that both $x$ and $y$ can be generated by a symbol we'll call $z$ which we'll think of as $x^{1/j}$, i.e. we'll give it the property that with $y=z^i$ and $x=z^j$. Then the map can be given as

$$\sum_{k,\ell=0}^{m,n}a_{k\ell}x^ky^\ell\mapsto \sum_{k,\ell=0}^{m,n}a_{k\ell}z^{kj+i\ell}$$